This is from Dummit and Foote (Section 9.2):
7. Determine all the ideals of the ring $\mathbb{Z} [x]/(2, (x^3+1))$.
This is my attempt to understand what's going on:
My plan is to find a nice list of cosets of $(2, x^3+1)$; hopefully once I do, I will be able to tell what kind of nice ring it is isomorphic to.
I wrote out the definition: $(2, (x^3+1)) = \{2p_1(x) + p_2(x) (x^3+1) |p_1(x), p_2(x) \in \mathbb{Z} [x]\}$. However, I cannot really get a grip on what this ideal looks like and what it's cosets are. So $2p_1(x)$ gives us all polynomials with even coefficients; $p_2(x) (x^3+1)$ gives us $some$ of the polynomials with some odd coefficients. All the polynomials from this term will have degree $\ge 3$. They will have a root at $x = -1$, and they also have $(x^2-x+1)$ as a factor.
How should I proceed?
Since $$ \mathbb{Z}[x]/(2, (x^3+1))\cong \mathbb{F}_2[x] /(x^3+1) ,$$
and $$ x^3+1=(x+1)(x^2+x+1) $$
in $ \mathbb{F}_2[x] $, and $ (x^2+x+1) $ is irreducible polynomial over $ \mathbb{F}_2 $.
In addition, every ideal in $ \mathbb{F}_2[x] /(x^3+1) $ corresponds to an ideal $ I $ of $ \mathbb{F}_2[x] $ which contains $ (x^3+1) $. That is $$ (x^3+1)\subset I .$$ Since $ \mathbb{F}_2[x] $ is a principal ideal domain, we can write $ I=(f(x)) $, where $ f(x)\in\mathbb{F}_2[x] $. Thus we have $$ (x^3+1)=(x+1)(x^2+x+1)\subset I=(f(x)) ,$$
which means $$ f(x)|(x+1)(x^2+x+1) $$ in $ \mathbb{F}_2[x] $.
So $ f(x)=1, (x+1), (x^2+x+1) $ or $(x^3+1) $ since $ \mathbb{F}_2[x] $ is unique factorization domain. And by corresponding theorem, ideals in $ \mathbb{F}_2[x]/(x^3+1) $ are $ (x+1), (x^2+x+1), (0), (1) $.